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<h1 id="Basic-Data-Structure-Representing-Meshes">Basic Data Structure Representing Meshes<a class="anchor-link" href="#Basic-Data-Structure-Representing-Meshes">&#182;</a></h1><h2 id="Data-structure:-node-and-elem">Data structure: node and elem<a class="anchor-link" href="#Data-structure:-node-and-elem">&#182;</a></h2><p>Two matrices <code>node(1:N,1:d)</code> and <code>elem(1:NT,1:d+1)</code> are used to represent a d-dimensional triangulation embedded in $\mathbb R^d$, where <code>N</code> is the number of vertices and <code>NT</code> is the number of elements.</p>
<p><code>node(k,1)</code> and <code>node(k,2)</code> are the x- and y-coordinates of the k-th node for points in 2-D. In 3-D, <code>node(k,3)</code> gives the additional z-coordinates of the k-th node.</p>
<p><code>elem(t,1:3)</code> are indices of 3 vertices of triangle <code>t</code>. <code>elem(t,1:4)</code> are indices of 4 vertices of tetrahedron <code>t</code>. By convention, the vertices are ordered such that the signed area/volume is positive. Therefore in 2-D, three vertices of a triangle is ordered counterclockwise and in 3-D, the ordering of four vertices of a tetrahedron follows the right-hand rule.</p>
<p>If the vertices are not positive ordering, call <code>elem = fixorder(node,elem)</code> to set all triangles counter-clockwise or <code>elem = fixorder3(node,elem)</code> to fix the orientation of tetrahedra.</p>
<p>For the following cases, a different ordering is used for other consideration.</p>
<ul>
<li>Edge and face elements; see <a href="sc3doc.html">Simplicial Complex in Three Dimensions</a></li>
<li><a href="uniformrefine3doc.html">Uniform refinement in 3-D</a> </li>
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<h2 id="Example:-L-shape-domain-in-2-D">Example: L-shape domain in 2-D<a class="anchor-link" href="#Example:-L-shape-domain-in-2-D">&#182;</a></h2>
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<div class=" highlight hl-matlab"><pre><span></span><span class="n">node</span> <span class="p">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">;</span> <span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">;</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">;</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">;</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span> <span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span> <span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">];</span>  <span class="c">% coordinates</span>
<span class="n">elem</span> <span class="p">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">8</span><span class="p">;</span> <span class="mi">3</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">2</span><span class="p">;</span> <span class="mi">8</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">;</span> <span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">3</span><span class="p">;</span> <span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">6</span><span class="p">;</span> <span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">8</span><span class="p">];</span>     <span class="c">% connectivity</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span> 
<span class="n">axis</span> <span class="n">on</span><span class="p">;</span>
<span class="n">findelem</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>  <span class="c">% plot indices of all triangles</span>
<span class="n">findnode</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>       <span class="c">% plot indices of all vertices</span>
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"
>
</div>

</div>

</div>
</div>

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<div class="text_cell_render border-box-sizing rendered_html">
<p>Apply the uniform refinement sevreal times to obtain a fine mesh.</p>

</div>
</div>
</div>
<div class="cell border-box-sizing code_cell rendered">
<div class="input">
<div class="prompt input_prompt">In&nbsp;[2]:</div>
<div class="inner_cell">
    <div class="input_area">
<div class=" highlight hl-matlab"><pre><span></span><span class="k">for</span> <span class="nb">i</span> <span class="p">=</span> <span class="mi">1</span><span class="p">:</span><span class="mi">3</span>
    <span class="p">[</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">]</span> <span class="p">=</span> <span class="n">uniformrefine</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="k">end</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
</pre></div>

</div>
</div>
</div>

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"
>
</div>

</div>

</div>
</div>

</div>
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
</div>
<div class="inner_cell">
<div class="text_cell_render border-box-sizing rendered_html">
<h2 id="Example:-A-Cube-in-3-D">Example: A Cube in 3-D<a class="anchor-link" href="#Example:-A-Cube-in-3-D">&#182;</a></h2>
</div>
</div>
</div>
<div class="cell border-box-sizing code_cell rendered">
<div class="input">
<div class="prompt input_prompt">In&nbsp;[3]:</div>
<div class="inner_cell">
    <div class="input_area">
<div class=" highlight hl-matlab"><pre><span></span><span class="n">node</span> <span class="p">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">;</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">];</span> 
<span class="n">elem</span> <span class="p">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">7</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">7</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">7</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">7</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">7</span><span class="p">];</span>
<span class="n">clf</span><span class="p">;</span> <span class="n">showmesh3</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,[],</span><span class="s">&#39;FaceAlpha&#39;</span><span class="p">,</span><span class="mf">0.25</span><span class="p">);</span>
<span class="n">view</span><span class="p">([</span><span class="o">-</span><span class="mi">53</span><span class="p">,</span><span class="mi">8</span><span class="p">]);</span>
<span class="n">axis</span> <span class="n">on</span>
<span class="n">findelem3</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">findnode3</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
</pre></div>

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<p>Unlike 2-D case, to apply uniform refinement to obtain a fine mesh with good mesh quality, a different ordering of the inital mesh, which may violate the positive ordering, should be used. See <a href="uniformrefine3doc.html">3 D Red Refinement</a>.</p>

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